Question

(a) Which of the following expresses the zero product property formally using quantifiers and variables? ∀ real numbers x and y, if xy = 0 then x = 0 and y = 0. ∀ real numbers x and y, if x = 0 and y = 0 then xy = 0. ∀ real numbers x and y, if x = 0 or y = 0 then xy = 0. ∀ real numbers x and y, if xy = 0 then x = 0 or y = 0. (b) Which of the following is the contrapositive of the zero product property? ∀ real numbers x and y, if x ≠ 0 or y ≠ 0 then xy ≠ 0. ∀ real numbers x and y, if xy ≠ 0 then x ≠ 0 or y ≠ 0. ∀ real numbers x and y, if x ≠ 0 and y ≠ 0 then xy ≠ 0. ∀ real numbers x and y, if xy ≠ 0 then x ≠ 0 and y ≠ 0. (c) Which of the following is an informal version (without quantifier symbols or variables) for the contrapositive of the zero product property? For any two real numbers, if at least one of them is nonzero then their product is nonzero. If the product of two real numbers is nonzero, then both numbers are nonzero. If the product of two real numbers is nonzero, then neither number is zero. If neither of two real numbers is zero, then their product is nonzero.

Answer 1

A) ∀ real numbers x and y, if xy = 0 then x = 0 or y = 0.

B) ∀ real numbers x and y, if xy ≠ 0 then x ≠ 0 and y ≠ 0.

c) If neither of two real numbers is zero, then their product is nonzero

Explanation:

A) Expression of zero product property using quantifiers and variables

∀ real numbers x and y, if xy = 0 then x = 0 or y = 0.

B) Contrapositive of the zero product property

∀ real numbers x and y, if xy ≠ 0 then x ≠ 0 and y ≠ 0.

C) The Informal version (without quantifier symbols or variables) for the contrapositive of the zero product property is : If neither of two real numbers is zero, then their product is nonzero